For whatever it's worth, the ENT novel
The Good That Men Do established that the last holdout to joining United Earth was, indeed, Australia -- the Independent Republic of Australia, which joined in 2150. The novel
Articles of the Federation had previously established that United Earth was founded in 2130 with the signing of the Traité d'Unification in Paris. Presumably, United Earth must have continued to accept new members until Australia closed the deal twenty years later.
It's clear that the dialogue in "Attached" does not
require Australia to have been the last holdout, but that's the interpretation the authors chose to go with.
What does this tell about political changes in Australia before the joining, BTW? Would the nation today be free to make decisions of that sort, or would it have to beg the Queen for permission?
Well, they wouldn't "beg" the Queen of Australia for "permission," because it's not hers to give. Political questions like that are the province of the Parliament and the Government of Australia, not the Queen, who must stay apolitical. Besides, the Queen's powers are exercised by the Governor General.
I've heard some arguments that any Government seriously proposing allowing Australia to be annexed by a foreign state -- supposedly the U.S. offered to accept Australia into the Union after World War II -- would by legal definition be considering treason against the Queen, however. I'm not sure if that's true or not -- or if allowing Australia to be annexed by United Earth would be the same thing. If that is the case, however, maybe the public would have passed a constitutional amendment allowing Australia to join United Earth (presuming Australia was still a monarchy rather than the republic the novels establish it to have been).
...So, how does it go? Would the death of Elizabeth II change something vis-á-vis Australia's ability to make decisions of this sort?
Well, not in terms of the constitutional functioning of the Commonwealth of Australia. Charles would ascend to the throne and become the new King of Australia, but he would be as obliged to stay politically neutral as his mother, and his powers would still be exercised by the Governor General. Constitutionally, the role of a hypothetical King Charles III of Australia would be no different from his mother's role as Queen Elizabeth II of Australia.
Would a new face beneath the crown make it likelier for Australia to cease to be ruled by the Queen/King of Australia, incidentally aka Queen/King of United Kingdom? Or is Australia going to go republic only after the UK does?
Well, Australian public opinion seems
quite divided. But it does seem as though support for the Australian Monarchy is at least in part contingent upon the popularity of the present Monarch. When she goes, that does look like it would help the cause of Australian republicanism. And it does seem as though an Australian republic is more likely than a British republic, especially since
the current Prime Minister has said Australia should become a republic after the Queen's death, and that the prior Prime Minister
refused to swear an Oath of Allegiance to the Queen when he was sworn in as PM by the Governor General.
The ENT finale appeared to describe the founding of a planetary alliance that was said to "give birth to" the Federation eventually, in a year that appeared to be 2160 or 2161. No episode of ENT prior to that suggested that a Federation would have been founded, though,
No, the Federation was explicitly referenced in the "Shockwave" two-parter and in Season Three's "Zero Hour," where Daniels brought Archer to the Federation's founding ceremony. The CGI set used at the climax of "These Are the Voyages..." was a reuse of the CGI set they used for the Federation's founding in "Zero Hour."
... why would Crusher pick 2150 for her example ...
"...
twenty one fifty? Would that have disqualified us as a Federation member?"
The year 2150 might have more to do with the process of Earth (and the others) becoming the founding members of the Federation, than anything involving Earth's uber-state and who joined when.
Except that earlier episodes had explicitly established the Federatin's founding at 2161. It's pretty clear that she's asking a hypothetical question about whether or not the rules of the eventual Federation would have had to have applied to a partially disunited Earth in order upon the Federation's founding, not trying to place the Federation's founding in 2150.
The creation of the Federation was likely a protracted multi-year series of events and procedures.
Depends on what you mean by "creation" and "the Federation." It's likely that the process which resulted in the Federation's establishment was a protracted, multi-year process involving prior legal entities, but that the establishment of the Federation itself has a specific date. (In the novels, the explicit date of the Federation's establishment is 12 August in 2161, with the ceremony itself being held at Candlestick Auditorium in San Francisco.)
To draw a comparison: The establishment of the United States was certainly a protracted, multi-year process. You have to start from the founding of the various colonies, to the legal evolution of the Thirteen Colonies, to the Declaration of Independence, to the establishment of the Articles of Confederation, to the establishment of the Constitution, to the constitutional evolution of the U.S. since the Constitution came into effect. Yet it's pretty clear that the actual state known as the United States of America came into existence on 21 June 1788. (The Declaration of Independence had declared each colony to be its own independent sovereign state -- its own country, in other words -- and the Articles had united them into an alliance more akin to the European Union today than a sovereign state in its own right.)
And we've seen part of the process of establishing the Federation, and it did indeed involve establishing a precursor entity: The Coalition of Planets, established in 2155. And we do indeed know that not all worlds who were involved in the establishment of the Coalition were involved in the Federation at first: The Rigel worlds, Denobula, and Coridan were involved in the early Coalition but not in the early Federation.